
Accession Number : AD0620162
Title : A GENERALIZATION OF THE CARTANKAHLER THEOREM.
Descriptive Note : Technical rept.,
Corporate Author : WASHINGTON UNIV SEATTLE DEPT OF MATHEMATICS
Personal Author(s) : Mansfield,Larry E.
Report Date : AUG 1965
Pagination or Media Count : 70
Abstract : An existence theorem for hyperbolic systems of partial differential equations is used to obtain a generalization of the CartanKahler theorem to nonanalytic differential systems. In order to apply this theorem certain restrictions must be placed on the differential system considered. A C(k)differential system S in r independent variables satisfying these conditions is said to be C(k)hyperbolic in the x(r)direction. After making this definition the following theorem is proved. Let S be C(k)hyperbolic in the x(r)direction with k > or = 4(r+1) and genus g > or = r. Suppose I sub (r1) is an (r1)dimensional C(k)integral submanifold of S. Then in a neighborhood of each regular point p contained in I sub (r1) of S there exists an rdimensional C(k)integral submanifold containing I sub (r1). In case r = 2 the differential system need only be C(1) for the theorem to hold. The concept of a moving frame is introduced and used to set up four applications of the above theorem to problems in surface theory. In the section on applications it is proved that locally a 2dimensional C(k)Riemannian manifold has a C(k)isometric imbedding in 3dimensional euclidean space for k > or = 1. It is shown that the theorem cannot be used to obtain a local C(k)conformal equivalence between two surfaces. (Author)
Descriptors : (*DIFFERENTIAL EQUATIONS, ALGEBRAIC GEOMETRY), SURFACES, GEOMETRY, INTEGRALS, PARTIAL DIFFERENTIAL EQUATIONS, THEOREMS
Distribution Statement : APPROVED FOR PUBLIC RELEASE